On Fano Indices of Q-fano 3-folds

We shall give the best possible upper bound of the Fano indices together with a characterization of those Q-Fano 3-folds which attain the maximum in terms of graded rings. 0. Introduction Q-Fano 3-folds play important roles in birational algebraic geometry. They have been studied by several authors since G. Fano. In this paper, we study Q-Fano 3-folds from the view of their Fano indices (See definition 0.2 below) and graded rings. More concretely, we give an optimal upper bound for the Fano indices and also characterize those Q-Fano 3-folds which attain the maximum in terms of graded rings (Theorem 0.3). Throughout this paper, we work over the complex number field C. Definition 0.1. Let X be a normal projective 3-fold. We call X a Q-Fano 3-fold if: (1) X has only Q-factorial terminal singularities; (2) the anti-canonical (Weil) divisor −KX is ample; and (3) ρ(X) = 1, ρ(X) is the Picard number of X. Let X be a Q-Fano 3-fold. There are two important indices of X: Definition 0.2. We define the Gorenstein index r = r(X) and the Fano inex f = f(X) by r(X) := min {n ∈ Z>0 |nKX is Cartier} ; f(X) := max {m ∈ Z>0 |KX = mA for some integral Weil divisor A} . Here the equality KX = mA means that KX −mA is linear equivalent to 0. If −KX = f(X)A, we call A = AX a primitive Weil divisor. In earlier works of Shokurov, Alexeev, Iskovskikh, Prokhorov, Sano, Mella and others the Fano index was defined in a different way, as the maximal rational such that −KX ≡ rH for some ample Cartier divisor H. Note that our definition is different from the one used by previous authors. Although the Gorenstein indices do not appear in the statement of main results, they play crucial roles in the proof (See section 2). Our main result is as follows: Theorem 0.3. Set F := {n ∈ Z>0|1 ≤ n ≤ 11, or 13, 17, 19} = {1, 2, · · · , 9, 10, 11, 13, 17, 19} . 1